The Annals of Applied Probability

Convergence rate for the approximation of the limit law of weakly interacting particles: application to the Burgers equation

Mireille Bossy and Denis Talay

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Abstract

In this paper we construct a stochastic particle method for the Burgers equation with a monotone initial condition; we prove that the convergence rate is $O(1/ \sqrt{N} + \sqrt{\Delta t})$ for the $L^1 (\mathbb{R} \times \Omega)$ norm of the error. To obtain that result, we link the PDE and the algorithm to a system of weakly interacting stochastic particles; the difficulty of the analysis comes from the discontinuity of the interaction kernel, which is equal to the Heaviside function.

In a previous paper we showed how the algorithm and the result extend to the case of nonmonotone initial conditions for the Burgers equation. We also treated the case of nonlinear PDE's related to particle systems with Lipschitz interaction kernels. Our next objective is to adapt our methodology to the (more difficult) case of the two-dimensional inviscid Navier-Stokes equation.

Article information

Source
Ann. Appl. Probab., Volume 6, Number 3 (1996), 818-861.

Dates
First available in Project Euclid: 18 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1034968229

Digital Object Identifier
doi:10.1214/aoap/1034968229

Mathematical Reviews number (MathSciNet)
MR1410117

Zentralblatt MATH identifier
0860.60038

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 65C20: Models, numerical methods [See also 68U20] 65M15: Error bounds 65U05

Keywords
Stochastic particle methods interacting particle systems Burgers equation

Citation

Bossy, Mireille; Talay, Denis. Convergence rate for the approximation of the limit law of weakly interacting particles: application to the Burgers equation. Ann. Appl. Probab. 6 (1996), no. 3, 818--861. doi:10.1214/aoap/1034968229. https://projecteuclid.org/euclid.aoap/1034968229


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